Let $\mathbb F$ be a finite field. A famous problem in the theory of polynomials over finite fields is the characterization of all nonconstant polynomials $F\in \mathbb F[x]$ for which the value set $\{F(\alpha): \alpha \in \mathbb F\}$ has the minimum possible size $\left\lfloor(\#\mathbb F-1)/\deg F \right\rfloor +1$. Such a polynomial is called an MVSP. The MVSPs are closely related to a special class of algebraic curves over finite fields, which are particularly interesting for applications. In this talk we will give an overview on the recent advances in the problem and discuss some of the machinery that has been used. Finally, we will briefly comment on a conjecture towards the solution of the problem that has been proposed by the author.

This is based on joint works with Herivelto Borges (ICMC-USP, Brazil).